On the irreducible solutions of the equation with inverses

Consider the following symmetric Diophantine equation
$$ \frac{1}{x_{1}}+\ldots + \frac{1}{x_{r}}\,=\,\frac{1}{x_{r+1}}+\ldots + \frac{1}{x_{2r}},\qquad (1) $$
where $r\geqslant 3$, and the variables $x_{1},\ldots, x_{2r}$ run through the segment $[1,N]$. Such equations appear in te problems connected with the estimates of incomplete Kloosterman sums.
The solution of (1) is called irreducible if any component from the set $x_{1},\ldots, x_{r}$ is not contained in the set $x_{r+1},\ldots, x_{2r}$. The following assertions holds true.
Theorem 1. Let $N,r\geqslant 3$. Then the number $J_{r}(N)$ of irreducible solutions of the equation (1) with positive integer variables $1\leqslant x_{1},\ldots, x_{2r}\leqslant N$ obeys the estimate:
$$ J_{r}(N)<e^{(3r)^{3}-90}N^{\,r\,-\,r/(2(2r-1))} \biggl(\frac{\ln{N}}{r}+9\biggr)^{\!10r^{2}}\!\!\exp{\biggl(\frac{26r^{3/2}\sqrt{\ln{N}}}{\ln{(r\ln{N})}}\biggr)}. $$

The estimate of Theorem 1 allows one to derive an asymptotic formula for the whole number $I_{r}(N)$ of solutions of the equation (1) with integer variables $1\leqslant x_{1},\ldots, x_{2r}\leqslant N$. Namely, we have
Theorem 2. Let $N,r\geqslant 3$. Then the number $I_{r}(N)$ satisfies the relation
$$ I_{r}(N)\,=\,r!N^{r}\bigl(1\,+\,\delta_{r}(N)\bigr), $$
where
$$ |\delta_{r}(N)|\leqslant e^{(3r)^{3}-90}N^{-\,r/(2(2r-1))}\biggl(\frac{\ln{N}}{r}+9\biggr)^{\!10r^{2}}\!\!\exp{\biggl(\frac{26r^{3/2}\sqrt{\ln{N}}}{\ln{(r\ln{N})}}\biggr)}. $$
In the talk, we briefly describe main ideas that allow one to derive the above theorems and some other assertions concerning the number of solutions of the equation (1).